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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,12,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 141.2
Character \(\chi\) \(=\) 230.141
Dual form 230.4.g.b.31.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.284630 + 1.97964i) q^{2} +(-1.76914 + 3.87387i) q^{3} +(-3.83797 + 1.12693i) q^{4} +(-3.27430 + 3.77875i) q^{5} +(-8.17242 - 2.39964i) q^{6} +(-26.5942 + 17.0910i) q^{7} +(-3.32332 - 7.27706i) q^{8} +(5.80424 + 6.69844i) q^{9} +(-8.41254 - 5.40641i) q^{10} +(2.31578 - 16.1066i) q^{11} +(2.42432 - 16.8615i) q^{12} +(-2.85088 - 1.83215i) q^{13} +(-41.4036 - 47.7823i) q^{14} +(-8.84568 - 19.3693i) q^{15} +(13.4601 - 8.65025i) q^{16} +(19.9821 + 5.86727i) q^{17} +(-11.6085 + 13.3969i) q^{18} +(52.6683 - 15.4648i) q^{19} +(8.30830 - 18.1926i) q^{20} +(-19.1597 - 133.259i) q^{21} +32.5444 q^{22} +(-77.0352 - 78.9467i) q^{23} +34.0698 q^{24} +(-3.55787 - 24.7455i) q^{25} +(2.81556 - 6.16522i) q^{26} +(-146.545 + 43.0295i) q^{27} +(82.8072 - 95.5646i) q^{28} +(-13.8771 - 4.07468i) q^{29} +(35.8266 - 23.0244i) q^{30} +(-3.19871 - 7.00419i) q^{31} +(20.9555 + 24.1840i) q^{32} +(58.2979 + 37.4658i) q^{33} +(-5.92760 + 41.2274i) q^{34} +(22.4947 - 156.454i) q^{35} +(-29.8252 - 19.1675i) q^{36} +(87.6813 + 101.190i) q^{37} +(45.6057 + 99.8627i) q^{38} +(12.1411 - 7.80262i) q^{39} +(38.3797 + 11.2693i) q^{40} +(177.057 - 204.335i) q^{41} +(258.351 - 75.8587i) q^{42} +(80.9235 - 177.198i) q^{43} +(9.26311 + 64.4264i) q^{44} -44.3166 q^{45} +(134.360 - 174.973i) q^{46} -374.802 q^{47} +(9.69726 + 67.4459i) q^{48} +(272.659 - 597.039i) q^{49} +(47.9746 - 14.0866i) q^{50} +(-58.0800 + 67.0279i) q^{51} +(13.0063 + 3.81900i) q^{52} +(-249.519 + 160.356i) q^{53} +(-126.894 - 277.859i) q^{54} +(53.2802 + 61.4886i) q^{55} +(212.753 + 136.728i) q^{56} +(-33.2688 + 231.389i) q^{57} +(4.11658 - 28.6315i) q^{58} +(-211.554 - 135.958i) q^{59} +(55.7774 + 64.3705i) q^{60} +(305.937 + 669.909i) q^{61} +(12.9554 - 8.32590i) q^{62} +(-268.842 - 78.9391i) q^{63} +(-41.9111 + 48.3680i) q^{64} +(16.2579 - 4.77375i) q^{65} +(-57.5755 + 126.073i) q^{66} +(-15.3596 - 106.828i) q^{67} -83.3026 q^{68} +(442.115 - 158.757i) q^{69} +316.125 q^{70} +(67.3543 + 468.460i) q^{71} +(29.4556 - 64.4988i) q^{72} +(-1137.06 + 333.871i) q^{73} +(-175.363 + 202.379i) q^{74} +(102.155 + 29.9955i) q^{75} +(-184.712 + 118.707i) q^{76} +(213.692 + 467.920i) q^{77} +(18.9021 + 21.8142i) q^{78} +(-1134.53 - 729.116i) q^{79} +(-11.3852 + 79.1857i) q^{80} +(58.5102 - 406.948i) q^{81} +(454.905 + 292.350i) q^{82} +(-358.620 - 413.870i) q^{83} +(223.707 + 489.851i) q^{84} +(-87.5983 + 56.2960i) q^{85} +(373.821 + 109.764i) q^{86} +(40.3353 - 46.5494i) q^{87} +(-124.905 + 36.6753i) q^{88} +(-74.3745 + 162.857i) q^{89} +(-12.6138 - 87.7310i) q^{90} +107.130 q^{91} +(384.626 + 216.182i) q^{92} +32.7923 q^{93} +(-106.680 - 741.975i) q^{94} +(-114.014 + 249.657i) q^{95} +(-130.759 + 38.3942i) q^{96} +(-673.651 + 777.434i) q^{97} +(1259.53 + 369.832i) q^{98} +(121.330 - 77.9743i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9} + 60 q^{10} - 51 q^{11} + 120 q^{12} + 184 q^{13} + 20 q^{14} - 15 q^{15} - 96 q^{16} - 334 q^{17} - 138 q^{18}+ \cdots - 10589 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{8}{11}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.284630 + 1.97964i 0.100632 + 0.699909i
\(3\) −1.76914 + 3.87387i −0.340470 + 0.745526i −0.999981 − 0.00616918i \(-0.998036\pi\)
0.659511 + 0.751695i \(0.270764\pi\)
\(4\) −3.83797 + 1.12693i −0.479746 + 0.140866i
\(5\) −3.27430 + 3.77875i −0.292863 + 0.337981i
\(6\) −8.17242 − 2.39964i −0.556063 − 0.163275i
\(7\) −26.5942 + 17.0910i −1.43595 + 0.922829i −0.436214 + 0.899843i \(0.643681\pi\)
−0.999736 + 0.0229855i \(0.992683\pi\)
\(8\) −3.32332 − 7.27706i −0.146871 − 0.321603i
\(9\) 5.80424 + 6.69844i 0.214972 + 0.248091i
\(10\) −8.41254 − 5.40641i −0.266028 − 0.170966i
\(11\) 2.31578 − 16.1066i 0.0634758 − 0.441484i −0.933156 − 0.359473i \(-0.882957\pi\)
0.996631 − 0.0820111i \(-0.0261343\pi\)
\(12\) 2.42432 − 16.8615i 0.0583200 − 0.405624i
\(13\) −2.85088 − 1.83215i −0.0608225 − 0.0390883i 0.509876 − 0.860248i \(-0.329691\pi\)
−0.570699 + 0.821160i \(0.693328\pi\)
\(14\) −41.4036 − 47.7823i −0.790399 − 0.912169i
\(15\) −8.84568 − 19.3693i −0.152263 − 0.333409i
\(16\) 13.4601 − 8.65025i 0.210313 − 0.135160i
\(17\) 19.9821 + 5.86727i 0.285080 + 0.0837071i 0.421146 − 0.906993i \(-0.361628\pi\)
−0.136066 + 0.990700i \(0.543446\pi\)
\(18\) −11.6085 + 13.3969i −0.152008 + 0.175427i
\(19\) 52.6683 − 15.4648i 0.635944 − 0.186730i 0.0521596 − 0.998639i \(-0.483390\pi\)
0.583784 + 0.811909i \(0.301571\pi\)
\(20\) 8.30830 − 18.1926i 0.0928896 − 0.203400i
\(21\) −19.1597 − 133.259i −0.199095 − 1.38473i
\(22\) 32.5444 0.315386
\(23\) −77.0352 − 78.9467i −0.698389 − 0.715718i
\(24\) 34.0698 0.289769
\(25\) −3.55787 − 24.7455i −0.0284630 − 0.197964i
\(26\) 2.81556 − 6.16522i 0.0212376 − 0.0465038i
\(27\) −146.545 + 43.0295i −1.04454 + 0.306705i
\(28\) 82.8072 − 95.5646i 0.558896 − 0.645001i
\(29\) −13.8771 − 4.07468i −0.0888591 − 0.0260914i 0.237001 − 0.971509i \(-0.423836\pi\)
−0.325860 + 0.945418i \(0.605654\pi\)
\(30\) 35.8266 − 23.0244i 0.218034 − 0.140122i
\(31\) −3.19871 − 7.00419i −0.0185324 − 0.0405803i 0.900139 − 0.435602i \(-0.143464\pi\)
−0.918672 + 0.395021i \(0.870737\pi\)
\(32\) 20.9555 + 24.1840i 0.115764 + 0.133599i
\(33\) 58.2979 + 37.4658i 0.307526 + 0.197635i
\(34\) −5.92760 + 41.2274i −0.0298993 + 0.207954i
\(35\) 22.4947 − 156.454i 0.108637 − 0.755586i
\(36\) −29.8252 − 19.1675i −0.138080 − 0.0887383i
\(37\) 87.6813 + 101.190i 0.389587 + 0.449608i 0.916334 − 0.400415i \(-0.131134\pi\)
−0.526747 + 0.850022i \(0.676588\pi\)
\(38\) 45.6057 + 99.8627i 0.194690 + 0.426312i
\(39\) 12.1411 − 7.80262i 0.0498496 − 0.0320364i
\(40\) 38.3797 + 11.2693i 0.151709 + 0.0445458i
\(41\) 177.057 − 204.335i 0.674431 − 0.778335i −0.310632 − 0.950530i \(-0.600541\pi\)
0.985063 + 0.172196i \(0.0550861\pi\)
\(42\) 258.351 − 75.8587i 0.949153 − 0.278696i
\(43\) 80.9235 − 177.198i 0.286993 − 0.628428i −0.710143 − 0.704058i \(-0.751370\pi\)
0.997136 + 0.0756301i \(0.0240968\pi\)
\(44\) 9.26311 + 64.4264i 0.0317379 + 0.220742i
\(45\) −44.3166 −0.146807
\(46\) 134.360 − 174.973i 0.430658 − 0.560833i
\(47\) −374.802 −1.16320 −0.581601 − 0.813474i \(-0.697574\pi\)
−0.581601 + 0.813474i \(0.697574\pi\)
\(48\) 9.69726 + 67.4459i 0.0291600 + 0.202812i
\(49\) 272.659 − 597.039i 0.794923 − 1.74064i
\(50\) 47.9746 − 14.0866i 0.135693 − 0.0398430i
\(51\) −58.0800 + 67.0279i −0.159467 + 0.184035i
\(52\) 13.0063 + 3.81900i 0.0346856 + 0.0101846i
\(53\) −249.519 + 160.356i −0.646681 + 0.415596i −0.822452 − 0.568835i \(-0.807394\pi\)
0.175771 + 0.984431i \(0.443758\pi\)
\(54\) −126.894 − 277.859i −0.319780 − 0.700220i
\(55\) 53.2802 + 61.4886i 0.130624 + 0.150748i
\(56\) 212.753 + 136.728i 0.507685 + 0.326269i
\(57\) −33.2688 + 231.389i −0.0773080 + 0.537689i
\(58\) 4.11658 − 28.6315i 0.00931955 − 0.0648189i
\(59\) −211.554 − 135.958i −0.466814 − 0.300003i 0.286008 − 0.958227i \(-0.407672\pi\)
−0.752822 + 0.658224i \(0.771308\pi\)
\(60\) 55.7774 + 64.3705i 0.120014 + 0.138503i
\(61\) 305.937 + 669.909i 0.642152 + 1.40612i 0.898259 + 0.439467i \(0.144833\pi\)
−0.256107 + 0.966648i \(0.582440\pi\)
\(62\) 12.9554 − 8.32590i 0.0265376 − 0.0170547i
\(63\) −268.842 − 78.9391i −0.537634 − 0.157863i
\(64\) −41.9111 + 48.3680i −0.0818576 + 0.0944687i
\(65\) 16.2579 − 4.77375i 0.0310238 − 0.00910940i
\(66\) −57.5755 + 126.073i −0.107380 + 0.235129i
\(67\) −15.3596 − 106.828i −0.0280071 − 0.194794i 0.971014 − 0.239021i \(-0.0768265\pi\)
−0.999021 + 0.0442274i \(0.985917\pi\)
\(68\) −83.3026 −0.148558
\(69\) 442.115 − 158.757i 0.771367 − 0.276986i
\(70\) 316.125 0.539774
\(71\) 67.3543 + 468.460i 0.112584 + 0.783041i 0.965390 + 0.260812i \(0.0839903\pi\)
−0.852805 + 0.522229i \(0.825101\pi\)
\(72\) 29.4556 − 64.4988i 0.0482136 − 0.105573i
\(73\) −1137.06 + 333.871i −1.82305 + 0.535297i −0.999490 − 0.0319274i \(-0.989835\pi\)
−0.823563 + 0.567224i \(0.808017\pi\)
\(74\) −175.363 + 202.379i −0.275480 + 0.317921i
\(75\) 102.155 + 29.9955i 0.157278 + 0.0461811i
\(76\) −184.712 + 118.707i −0.278788 + 0.179166i
\(77\) 213.692 + 467.920i 0.316266 + 0.692525i
\(78\) 18.9021 + 21.8142i 0.0274390 + 0.0316663i
\(79\) −1134.53 − 729.116i −1.61575 − 1.03838i −0.958655 − 0.284572i \(-0.908148\pi\)
−0.657096 − 0.753807i \(-0.728215\pi\)
\(80\) −11.3852 + 79.1857i −0.0159113 + 0.110665i
\(81\) 58.5102 − 406.948i 0.0802609 − 0.558227i
\(82\) 454.905 + 292.350i 0.612633 + 0.393715i
\(83\) −358.620 − 413.870i −0.474261 − 0.547327i 0.467331 − 0.884083i \(-0.345216\pi\)
−0.941592 + 0.336756i \(0.890670\pi\)
\(84\) 223.707 + 489.851i 0.290577 + 0.636275i
\(85\) −87.5983 + 56.2960i −0.111781 + 0.0718372i
\(86\) 373.821 + 109.764i 0.468723 + 0.137630i
\(87\) 40.3353 − 46.5494i 0.0497057 − 0.0573634i
\(88\) −124.905 + 36.6753i −0.151305 + 0.0444273i
\(89\) −74.3745 + 162.857i −0.0885807 + 0.193965i −0.948738 − 0.316065i \(-0.897638\pi\)
0.860157 + 0.510030i \(0.170366\pi\)
\(90\) −12.6138 − 87.7310i −0.0147735 − 0.102752i
\(91\) 107.130 0.123410
\(92\) 384.626 + 216.182i 0.435870 + 0.244984i
\(93\) 32.7923 0.0365634
\(94\) −106.680 − 741.975i −0.117055 − 0.814137i
\(95\) −114.014 + 249.657i −0.123133 + 0.269624i
\(96\) −130.759 + 38.3942i −0.139016 + 0.0408187i
\(97\) −673.651 + 777.434i −0.705143 + 0.813778i −0.989438 − 0.144959i \(-0.953695\pi\)
0.284295 + 0.958737i \(0.408241\pi\)
\(98\) 1259.53 + 369.832i 1.29828 + 0.381211i
\(99\) 121.330 − 77.9743i 0.123173 − 0.0791587i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 230.4.g.b.141.2 yes 60
23.8 even 11 inner 230.4.g.b.31.2 ✓ 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.b.31.2 ✓ 60 23.8 even 11 inner
230.4.g.b.141.2 yes 60 1.1 even 1 trivial